# Spatiotemporal Intermittency in Pulsatile Pipe Flow

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Numerical Methodology

#### 2.1. Governing Equations

#### 2.2. Direct Numerical Simulation

`[22]. In`

**nsPipe**`, the governing Equation (2) are treated in cylindrical coordinates $\left(\right)$ and discretised using a Fourier–Galerkin ansatz in $\theta $ and z and high-order finite differences in r. No-slip boundary conditions are imposed at the solid pipe wall and periodic boundary conditions in $\theta $ and z. The discretised NSEs are integrated forward in time using a second-order predictor–corrector method with variable time-step size; details are given in López et al. [22] and the references therein. We have modified`

**nsPipe**`to account for a time-dependent driving force that maintains a pulsating flow rate according to Equation (1) and an additional volume force ${F}_{p}$ to perturb the flow locally.`

**nsPipe**#### 2.3. Transient Growth Analysis

#### 2.4. Modelling Geometric Imperfections in Our DNS

## 3. Results

#### 3.1. Temporal Modulation of Statistically Steady Puff Dynamics

#### 3.2. Optimal Infinitesimal Perturbations of Pulsatile Pipe Flow

#### 3.3. Nonlinear Dynamics of Helical Perturbations

#### 3.4. Puff Recovery Length

#### 3.5. Intermittent Production and Dissipation

#### 3.6. Effect of Local Geometric Imperfections

## 4. Discussion and Conclusions

## Author Contributions

## Funding

`. Computational resources were provided by HLRN through the project`

**Instabilities, Bifurcations and Migration in Pulsating Flow (FOR 2688)**`hbi00041`and are also gratefully acknowledged.

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

AC | Acceleration |

DC | Deceleration |

SW | Sexl–Womersley |

NSE | Navier–Stokes equations |

TGA | Transient growth analysis |

DNS | Direct numerical simulation |

SSPF | Statistically steady pipe flow |

IC SSPF | Cases with a SSPF initial condition |

IC SWOP | Cases with a SW profile and optimum perturbation initial condition |

## References

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**Figure 1.**Instantaneous representation of localised turbulent structures in a statistically steady pipe flow ($R\phantom{\rule{-1.00006pt}{0ex}}e=2400$, $A=0.0$). Grey surfaces represent low-speed streaks (${u}_{z}^{\prime}=-0.4{u}_{b}$) and blue/red surfaces represent positive/negative axial vorticity (${\omega}_{z}=\pm 6\raisebox{1ex}{${u}_{b}$}\!\left/ \!\raisebox{-1ex}{$D$}\right.$). (

**a**) Puff splitting. (

**b**) Single puff. (

**c**) Weak slug. The exact location and time for each snapshot are indicated in Figure 2a. The direction of the mean bulk flow (${u}_{s}$) is always from left to right.

**Figure 2.**Spatiotemporal representation of the turbulence activity in the computational pipe domain based on the cross-sectional average of the streamwise vorticity (${\omega}_{z}$) plotted on a logarithmic scale and in a co-moving reference frame. Steady ($A=0$,

**a**) and pulsatile (

**b**–

**f**) pipe flow at $R\phantom{\rule{-1.00006pt}{0ex}}e=2400$, $W\phantom{\rule{-1.00006pt}{0ex}}o=8$, and different amplitudes A. Initial conditions for all $A\ne 0$ were either taken from the steady case at time $\raisebox{1ex}{$t$}\!\left/ \!\raisebox{-1ex}{$T$}\right.=0.25$ (

**b**–

**d**,

**f**) or composed of a localised helical perturbation on top of the laminar Sexl–Womersley velocity profile (

**e**).

**Figure 3.**Sexl–Womersley (SW) flow and its optimal perturbation for ($R\phantom{\rule{-1.00006pt}{0ex}}e=2400$, $W\phantom{\rule{-1.00006pt}{0ex}}o=8$, $A=1.0$). (

**a**) Time-dependent velocity profile (${u}_{\mathrm{SW}}$) for 20 equispaced points within one pulsation period (T). Circles denote the maximum and minimum peak flow (PF), whereas upward- and downward-facing triangles denote phases of acceleration (AC) and deceleration (DC), respectively. (

**b**) Optimal helical perturbation during DC ($\raisebox{1ex}{$t$}\!\left/ \!\raisebox{-1ex}{$T$}\right.=0.2$) according to our transient growth analysis based on the linearised Navier–Stokes equations. To be used as initial condition in our direct numerical simulation (DNS) (Section 3.3), the helix is scaled to an amplitude of 4 × 10

^{−2}${u}_{s}$. (

**c**) Evolution of the optimal perturbation under the constraints of the linearised Navier–Stokes equations at the later time of maximal energy amplification. Note that, in the framework of transient growth analysis, the absolute amplitude of the initial helix is not important; only the relative growth rate is of interest. The dashed lines correspond to the Stoke layer thickness ($\delta $).

**Figure 4.**Geometric representation of the perturbation force (${\mathit{F}}_{p}$) in terms of iso-surfaces (black) of the localisation function for ${f}_{p}=0.5$. (

**a**) Axisymmetric contraction. (

**b**) Localised bump. (

**c**) Tilted bump. See Table 1 for details. The direction of the mean bulk flow (${u}_{s}$) is always from left to right.

**Figure 5.**Turbulent fraction (${F}_{t}$) in the computational pipe domain based on the axial vorticity data shown in Figure 2 and Figure 7. The threshold to distinguish turbulent from laminar regions is set to ${\langle {\omega}_{z}^{2}\rangle}_{r,\theta}=4\times {10}^{-2}$. (

**a**) Time series of the turbulent fraction for several amplitudes A (line styles) and different numerical set-ups (symbols and colours from those in (

**b**)). (

**b**) Time-averaged turbulent fraction ${\langle {F}_{t}\rangle}_{t>2}$ for four different set-ups: The statistically steady pipe flow (SSPF) serves as reference data and as initial condition (IC) for the first set-up. The IC for the second set-up are composed out of the analytical Sexl–Womersley (SW) velocity profile superimposed with an optimal perturbation (OP). The third set-up is initialised with an unperturbed SW flow and then permanently perturbed using a localised body force (see Section 3.6).

**Figure 6.**Instantaneous representation of localised turbulent structures in a pulsatile pipe flow ($R\phantom{\rule{-1.00006pt}{0ex}}e=2400$, $W\phantom{\rule{-1.00006pt}{0ex}}o=8$, $A=0.5$). Grey surfaces represent low-speed streaks (${u}_{z}^{\prime}=-0.4$${u}_{s}$) and blue/red surfaces represent positive/negative axial vorticity (${\omega}_{z}=\pm 8\raisebox{1ex}{${u}_{s}$}\!\left/ \!\raisebox{-1ex}{$D$}\right.$). (

**a**) Death of downstream puff. (

**b**) Splitting event. (

**c**) Growing puff. (

**d**) Isolated puff. The exact location and time for each snapshot are as indicated in Figure 2d. The direction of the mean bulk flow (${u}_{s}$) is always from left to right.

**Figure 7.**Spatiotemporal representation of the turbulence activity in the pipe domain based on the cross-sectional average of the streamwise vorticity (${\omega}_{z}$) plotted on a logarithmic scale and in a co-moving reference frame (${z}^{*}$). For pulsatile pipe flow at ($R\phantom{\rule{-1.00006pt}{0ex}}e=2400$, $W\phantom{\rule{-1.00006pt}{0ex}}o=8$). (

**a**–

**e**) For different pulsation amplitudes A, always using the SWOP initial condition. Note that the optimal time of perturbation slightly changes with A. The horizontal straight lines mark regions for which three-dimensional representations of the localised flow structures are shown in Figure 8. (

**f**) For a permanent body force and the unperturbed SW velocity profile as initial condition. The curved black line represents the fixed location of the highly localised body force viewed from the co-moving reference frame. The direction of the mean bulk flow (${u}_{s}$) is always from left to right.

**Figure 8.**Instantaneous representation of localised turbulent structures in a pulsatile pipe flow DNS at $R\phantom{\rule{-1.00006pt}{0ex}}e=2400$, $W\phantom{\rule{-1.00006pt}{0ex}}o=8$, and two different amplitudes. (

**a**–

**d**) Growth and decay of an initial helix at $A=1.0$. (

**e**–

**h**) Development of a puff at $A=0.5$. Both DNS were initialised at $\raisebox{1ex}{$t$}\!\left/ \!\raisebox{-1ex}{$T$}\right.=0.2$ using the SWOP initial condition. Grey surfaces represent low-speed streaks (${u}_{z}^{\prime}=-0.4$${u}_{s}$) and blue/red surfaces represent positive/negative axial vorticity (${\omega}_{z}=\pm 8\raisebox{1ex}{${u}_{s}$}\!\left/ \!\raisebox{-1ex}{$D$}\right.$). The exact location for each snapshot is as indicated in Figure 2e and Figure 7c, respectively. (

**a**) Decay. (

**b**) Breakdown into turbulence. (

**c**) Amplification of helix. (

**d**) Localised optimal helix perturbation. (

**e**,

**f**) Birth of a downstream puff. (

**g**) Amplification of helix. (

**h**) Localised optimal helix perturbation. Note that the initial perturbation is two orders of magnitude smaller. The direction of the mean bulk flow (${u}_{s}$) is always from left to right.

**Figure 9.**Instantaneous streamwise velocity profiles (${u}_{z}$) at five axial locations along the pipe for an IC SWOP simulation at $R\phantom{\rule{-1.00006pt}{0ex}}e=2400$, $W\phantom{\rule{-1.00006pt}{0ex}}o=8$, and $A=0.5$. To not interfere with one another, they are scaled in arbitrary physical units, since, in this representation, only the development in time and deviation from the SW profile are of interest. Thus, the velocity is scaled so its all-time maximum ${u}_{z}\left(\right)open="("\; close=")">r,\theta =0,z$ is equal to $10D$. Each profile is compared with the corresponding instantaneous SW profile (grey lines, also scaled) and its inflection point (grey circles) if they fulfil the Fjortoft criterion. The shaded grey area shows the instantaneous cross-sectional average of the streamwise vorticity (${\langle {\omega}_{z}^{2}\rangle}_{r,\theta}$) scaled so its all-time maximum is equal to $0.5D$.

**Figure 10.**Production (

**a**) and dissipation (

**b**) of turbulent kinetic energy compared for different phases of the pulsation period for $A=0.6$ using the SWOP initial conditions. Averages are taken over space- and phase-logged time instants ($\alpha =\theta ,z,\varphi $) over four periods of puff dynamics, excluding the initial period without puffs. Circles denote the existence and wall-normal location of the inflection points of the corresponding mean profile ${\partial}^{2}{\langle {u}_{z}\rangle}_{\varphi ,\theta ,z}/{\partial}^{2}r=0$ that satisfy the Fjortoft criterion. The vertical dashed line denotes the Stokes layer.

**Figure 11.**Production (

**a**) and dissipation (

**b**) of turbulent kinetic energy compared for different phases of the initial pulsation period for $A=1$ using the SWOP initial conditions.

**Figure 12.**Spatiotemporal representation of the turbulence activity in the pipe domain based on the cross-sectional average of the streamwise vorticity (${\omega}_{z}$) plotted on a logarithmic scale and in a stationary reference frame. Pulsatile pipe flow at $R\phantom{\rule{-1.00006pt}{0ex}}e=2400$, $W\phantom{\rule{-1.00006pt}{0ex}}o=8$, and different amplitudes A. Initial conditions are based on the Sexl–Womersley velocity profile, and there is a permanent body force. (

**a**–

**c**) Local bump. (

**d**) Tilted bump.

**Figure 13.**Instantaneous representation of localised turbulent structures in a pulsatile pipe flow DNS at ($R\phantom{\rule{-1.00006pt}{0ex}}e=2400$, $W\phantom{\rule{-1.00006pt}{0ex}}o=8$, $A=1.4$). The DNS was initialised at $\raisebox{1ex}{$t$}\!\left/ \!\raisebox{-1ex}{$T$}\right.=0.25$ using the corresponding SW profile and by introducing a local bump like body force, as described by Equation (3) and Table 1. Grey surfaces represent low-speed streaks (${u}_{z}^{\prime}=-0.2$ ${u}_{s}$) and blue/red surfaces represent positive/negative axial vorticity (${\omega}_{z}=\pm 2\raisebox{1ex}{${u}_{s}$}\!\left/ \!\raisebox{-1ex}{$D$}\right.$ for all panels except (

**d**) and (

**h**). There, it is $\pm 0.8\raisebox{1ex}{${u}_{s}$}\!\left/ \!\raisebox{-1ex}{$D$}\right.$. (

**a**–

**d**) Local bump. (

**e**–

**h**) Tilted bump. The exact instants in time are given in Figure 12c,d. The direction of the mean bulk flow (${u}_{s}$) is always from left to right.

**Table 1.**Parameters to control the body force term in Equation (3) to model the effect of geometric perturbations: Magnitude (${A}_{p}$) and slope (M), size (L), and location in the radial (r), azimuthal ($\theta $), and axial (z) direction. Geometric representations of the perturbations are shown in Figure 4.

${\mathit{A}}_{\mathit{p}}$ | ${\mathit{M}}_{\mathit{z}}$in$\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$\mathit{D}$}\right.$ | ${\mathit{L}}_{\mathit{z}}$inD | ${\mathit{z}}_{0}$inD | ${\mathit{M}}_{\mathit{r}}$ in $\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$\mathit{D}$}\right.$ | ${\mathit{r}}_{0}$inD | ${\mathit{M}}_{\mathit{\theta}}$ | ${\mathit{L}}_{\mathit{\theta}}$ | $\mathbf{\Delta}\mathit{\theta}$ | |
---|---|---|---|---|---|---|---|---|---|

Contraction | 0.25 | 4 | 2.5 | 10 | 100 | 0.45 | 20 | ≥1 | 0 |

Bump | 0.25 | 4 | 2.5 | 10 | 100 | 0.45 | 20 | 0.25 | 0 |

Tilted Bump | 0.25 | 4 | 2.5 | 10 | 100 | 0.45 | 20 | 0.0625 | 0.1 |

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**MDPI and ACS Style**

Feldmann, D.; Morón, D.; Avila, M.
Spatiotemporal Intermittency in Pulsatile Pipe Flow. *Entropy* **2021**, *23*, 46.
https://doi.org/10.3390/e23010046

**AMA Style**

Feldmann D, Morón D, Avila M.
Spatiotemporal Intermittency in Pulsatile Pipe Flow. *Entropy*. 2021; 23(1):46.
https://doi.org/10.3390/e23010046

**Chicago/Turabian Style**

Feldmann, Daniel, Daniel Morón, and Marc Avila.
2021. "Spatiotemporal Intermittency in Pulsatile Pipe Flow" *Entropy* 23, no. 1: 46.
https://doi.org/10.3390/e23010046